So the names are tied to group theory, and refer to the set of things you can do that make the soup look like it started. C1 is the trivial group, the only thing you can do is nothing. C2 and C4 are groups with 2 and 4 steps, rotations by a half or quarter circle (C means cyclic). So a C2 soup looks the same when rotated 180 degrees. A C4 soup looks the same when rotated 90 degrees. But because we are on a grid, there are three different alignments of the 180 degree rotated piece. The extra number tells that: for C2_2 a cell and its rotated image are in corners of a odd by even rectangle, C2_4 an even by even, and C2_1 an odd by odd.

The D is for the dihedral group. It's like the cyclic group but with reflections too. D2 is like C1 with a mirror down the middle, and again there is a choice of even or odd. There is an additional choice though, of whether the mirror is vertical/horizontal or diagonal. If the mirror is vertical/horizontal, it is +, for diagonal it is x. So D2 +2 refers to a soup that can be made to look like it started either by doing nothing or by reflecting along a horizontal line with an even distance between similar cells. Similarly, D4 resembles C2, but with the reflections too, and now there are 2 independent parities, one for the spin and one for the flip. Thus with D4, 1 means odd odd, 2 odd even, and 4 even even. So D4 soups look the same if you do nothing, if you spin 180, if you mirror appropriately or if you mirror and spin (4 possible operations, so the group is D4, which is the method behind identifying the number immediately following the letter).

D8 has the most options, it refers to soups that are identical if rotated by any multiple of 90 degrees or if flipped appropriately about any axis. A flip about a horizontal axis followed by a 90 degree rotation is the same as a flip about a diagonal axis so there's no + vs x in D8, only the even or odd.

You can tell the symmetry of an object with the same group theory. If some ash object can be mirrored about a line with an odd distance between neighbors (e.g., MWSS on MWSS 1), then it is probably more common from soups with D2_1 symmetry (which includes D4_1, D4_2, and D8_1 but not D4_4). 28p7.2, for another example, has C2_2 symmetry, so our best bet for finding a p7 from a soup would be to modify apgnano to support C2_2 -nudge nudge-.

Hope this helps! I started by writing from memory but I went and checked with catagolue and found I had a number of things wrong with the parity numbers, so I edited that all up and now it's fixed. Hopefully nobody saw it while it was still wrong..

SuperSupermario24 wrote:Speaking of symmetries, I have a couple questions about rule table symmetries:-What does the "permute" symmetry mean?-What is "rotate8"? How would that work with a Moore neighborhood?